A Path Loss Model for Non-Line-of-Sight Ultraviolet Multiple Scattering Channels

In free space optical communication, the deep ultraviolet (UV) spectrum with wavelength 200 280 nm is regarded as an appealing choice to overcome solar background radiation and relax pointing and tracking [3]. High altitude ozone absorbs most solar radiation in this band, yielding negligible background noise at sea level [4]. And, atmospheric scattering is very strong, enabling non-line-of-sight (NLOS) communications where the transmitter is not necessarily within the receiver field of view (FOV). However, in addition to scattering, strong atmospheric absorption leads to significant signal attenuation and so limits achievable rates.

These properties have motivated development of UV signal propagation models and communication systems for military and civilian applications, for example, long-range communication based on high-power UV lasers since the 1960s [5‐8]. Recent progress in deep UV light emitting diodes (LEDs) [9] and avalanche photodiodes (APDs) [10, 11] offers promise for deployment of low-cost/power and moderate bandwidth short-range UV communication systems [12, 13], including underwater communications [14, 15] and sensor networks [16, 17].

NLOS channel modeling is more complex than traditional LOS links. In addition to wavelength and device characteristics, NLOS path loss is a function of system geometry, including transmitter (Tx) beamwidth, communication range, receiver (Rx) FOV, the pointing elevation angles, as well as the optical properties of the atmosphere.

For simplicity and tractable analysis, single scattering models for NLOS communication links were developed [1, 18], with a corresponding single scattering assumption imposed; that is, each photon undergoes only a single interaction with the atmosphere before it reaches the detector. Recently the model has been further simplified for improved analytical tractability [19]. However, the single scattering assumption does not always lead to accurate link performance prediction, especially as the range increases, and with beam pointing at low elevation angles. Multiple scattering may occur when the particle density is large and/or the propagation distance is long. Alternatives to the single scattering model include an empirical path loss model [20] and a Monte Carlo statistical path loss model [2]. Those models are applicable for predicting path loss in a variety of scenarios and are generally more accurate than the single scattering model.

In this paper, following the same physical scattering law as the Monte Carlo statistical method [2, 21], we develop a stochastic analytical NLOS UV channel path loss model. We apply a stochastic analytical technique [22, 23] to theoretically derive the th scattered signal energy collected by the detector. The model assumes that the photons are stochastically scattered and/or absorbed by the atmospheric particles and involves probabilistic modeling of photon random moving direction, distance, energy loss, and receiver capture after a specified number of scatterings. In order to obtain the th-order scattered signal at the receiver, we trace the migration routes of a single photon through the medium. Its scattering distance and scattering angles follow certain probability distribution functions (PDFs) [24]. The propagation of a photon between two consecutive scatterers is modeled as a single scattering event. The probability of the photon arriving at the receiver is a function of the scattering events encountered. A similar technique has been applied to model multiply scattered lidar returns in cloudy media where significant scattering occurs [22]. To account for a divergent UV beam profile, the contributions of photons at all possible directions within the beam are integrated [25].

We assume a homogeneous atmosphere with constant scattering and absorption coefficients and ignore atmospheric turbulence. This assumes that different types of particles are well mixed and the environment is stationary. Photons are scattered elastically which conserves energy but incurs energy loss during propagation. The detector is assumed small in size compared with the propagation space and is regarded as a point detector with finite FOV.

Our model offers an analytical formulation for NLOS scattering channel path loss and provides a reference to easily check other models in a variety of system parameter settings. We consider the effects of different optical pointing geometries and find that path loss is relatively insensitive to the Tx beam angle but increases considerably with increased Tx/Rx elevation angles and decreased Rx FOV for medium communication range. Not surprisingly, numerical tests demonstrate good agreement with Monte Carlo simulation [2]. However, the current analytical approach can distinguish the contributions of different orders of multiple scattering, whereas Monte Carlo modeling can only capture the total scattering effect. A similar comparison of Monte Carlo simulation and analytical methods has also been described by Lavigne et al. [26].

The model reveals when high-order scattering plays a role, and thus offers more insight into the channel behavior. Extensive numerical tests show that high-order scattering may play a significant role for scenarios with large baseline ranges and elevation angles. Atmospheric conditions also affect the multiple scattering contributions. A tenuous atmosphere generally leads to higher path loss at a small range than a thick or extra thick atmosphere where rich multiple scattering enhances received signal strength. As a by-product, when contributions from only single scattering are retained, the corresponding single scattering path loss shows a good match with that predicted by Reilly's analytical single scattering model [1]. A disadvantage of the proposed model is that it cannot provide pulse spreading information, while the Monte Carlo model can [2].

The organization of this paper is as follows. The stochastic path loss model based on a general NLOS UV channel configuration is developed in Section 2. It consists of modeling photon direction, distance traveled, probability of arrival at the receiver after scattering times, and total path loss. Numerical analysis for the multiple scattering NLOS channel is carried out and path loss performance is compared with both the single scattering model and Monte Carlo simulation in Section 3. Channel characteristics under different geometric parameters are analyzed in Section 4. Finally some conclusions are drawn.

A NLOS UV channel involves rich scattering and absorption because of abundant suspended particulates in the atmosphere. For NLOS UV communication, scattering serves as the vehicle for information exchange between the Tx and Rx. The scattered photons that reach the receiver depend on the link geometry, the atmospheric optical properties, and their random migration, as described next.

In this section we study modeling performance in terms of the received energy ratio and path loss results for different geometry scenarios. Through numerical simulation, multiple scattering effects are observed and analyzed. These results are also compared with Reilly's analytical single scattering model [1] as well as the Monte Carlo simulated multiple scattering model [2], respectively, under the same geometries and atmosphere parameters. We demonstrate good agreement with [1] when single scattering is of concern, and with [2] when multiple scatterings are considered.

We assume gas concentrations and optical features of the atmosphere described by [7, Table II] for middle UV at wavelength 260 nm. Large dynamic ranges for the absorption and Mie scattering coefficients indicate that weather conditions may significantly affect the UV signal propagation. However, an explicit correspondence between the parameter settings and weather conditions is not available in the literature. Therefore we consider atmosphere coefficients for typical tenuous, thick, and extra thick atmosphere conditions (corresponding to clear, overcast, and foggy), given in Table 1. The tenuous condition will be adopted for all the following results unless stated otherwise. This agrees with the experimental measurement conditions in [13]. We set the geometric and model parameters as follows: ( ) = (17°, 30°, 60°, 60°), ranges from 10 m to 1000 m, , , , and the detector area is . Unless otherwise specified, the same parameters are used in Section 4.

To illustrate the importance of multiple scattering, consider the following example. Figure 3 shows the ratio of the accumulated received energy from (1st scattering order only) to (sum of 1st and 2nd-order scatterings), and (sum of 1st, 2nd and 3rd order scatterings), all predicted via our proposed stochastic modeling for different optical geometries. We observe that, for each scenario, the 2nd-order scattering contributes little to the total received energy for ranges within 100 meters, but it will increasingly contribute more with longer ranges and larger pointing angles. Further, considering does not add much to the received energy. For this case, 2nd-order scattering is sufficient because 3rd order scattering makes only a negligible contribution to the received energy.

The received signal energy can be transformed to the link path loss in decibels, depicted for different geometries in Figure 4. We also compare the proposed model with Reilly's single scattering model. Both models predict similar path loss levels for . As expected, including multiple scattering reduces predicted path loss. The improvement is not obvious for short range, consistent with Figure 3. Again, the 3rd-order scattering does not lead to a significant improvement for the received energy for these cases.

Due to the fact that our proposed stochastic model has the same premise that underlies the Monte Carlo simulation, we should expect the similar path loss from both models. As an example, Figure 5 compares the path loss results generated from our stochastic model and the Monte Carlo simulation model. Four sets of Tx and Rx elevation angles are included for detailed comparison. For each subplot, the stochastic model provides a good match with the simulation model within the high precision numerical accuracy of the simulation. This indicates that our proposed model is a good reference for evaluating the Monte Carlo simulation model.

Since NLOS UV channel characteristics are crucial to communication system design, in this section, we apply the proposed multiple scattering model to further study the effects of different system parameters on the channel path loss, including link geometries and two typical atmosphere conditions. Referring to Figure 1, we study angle sensitivity by varying one angle and keeping the others fixed.

Figure 6 illustrates range-dependent path losses for (a) varying Tx elevation angle , (b) varying Rx elevation angle , (c) varying Tx beam angle , and (d) varying Rx FOV angle . Note that path loss is not very sensitive to the Tx beam angle. By contrast, path loss has the heaviest dependence on the Rx FOV angle. A wider FOV enables collection of more scattered photons which results in lower path loss. Path loss varies slightly with Rx elevation angle and moderately with Tx elevation angle. Larger baseline range tends to make the angle effects more pronounced. As an example, in the first subplot for baseline range 1000 m, path losses for three different Tx elevation angles show differences up to 8 dB, while for a shorter range of  m, the difference is within  dB. Similar effects are obtained for other angular variation. One possible reason is that the roles of scattering and absorption may vary for different geometries. Larger propagation distance may bring about more scattered photons to the receiver but will also lead to higher absorption loss.

Up to now we have only considered tenuous atmospheric conditions. Next we consider the impact of atmospheric conditions on path loss. Figure 7 depicts range-dependent path loss for tenuous, thick, and extra thick atmosphere conditions whose coefficients are given in Table 1. Four pairs of Tx and Rx elevation angles are incorporated for performance analysis. For elevation angle pair ( , ), path loss results for extra thick atmosphere are about 15 dB less than for tenuous, and  dB smaller than thick atmosphere at  m. As the range increases, path loss increases more dramatically. The gaps between curves decrease until a few hundred meters. Further range increase leads to a dramatic path loss increase for extra thick atmosphere. As elevation angles increase, the lower path loss region tends to shift to a smaller range. For example, for the medium elevation angle pairs (45°, 45°) and (60°, 60°), the lower attenuation range shrinks to less than 100 m, and for elevation angle pair (90°, 90°), it reduces to less than  m. In these scenarios, the scattering contribution dominates absorption loss for short ranges. For larger baseline ranges, scattering loss increases as the atmosphere gradually becomes thick and thus causes large path loss. For a given atmospheric condition, path loss increases by  dB to  dB when the elevation angle increases from to 90° at short range. These observations and predictions are helpful for experimental channel characterization and communication system design.

In order to further demonstrate the significance of high-order scatterings, in Figure 8 we plot the ratio of the accumulated received energy from (1 -order scattering only) to (sum of 1st through 5th-order scatterings) for an extra thick atmosphere at different elevation angle pairs. For each pointing scenario, we observe that the 1st-order scattering contributes dominantly to the total received energy for short ranges up to 20 m. As the range increases, high-order scattering will contribute more to the total received energy. For example, the 2nd-, 3rd-, and 4th-order scattering overwhelm other order scattering for elevation angle (90°, 90°) at a range of  m. Further, we can see that the 5 -order scattering does not contribute as much as the lower order scatterings. We also depict the corresponding path loss in Figure 9. When considering multiple scattering contributions, the path loss decreases as higher order scattering contributions are added. When compared with tenuous atmosphere, an extra thick atmosphere yields significant high-order scattering contributions, such as 3rd- and 4th-order scatterings, which should be accounted for when predicting path loss. These results indicate the remarkable property that for relatively shorter ranges, NLOS UV communications links are enhanced in hazy or foggy weather.

This paper proposed an analytical energy loss model for NLOS UV communication channels based on the probability theory of scattering and absorption. The model was developed by employing the PDFs of scattering distance and scattering angles. Multiple scattering was incorporated and contributions of different scattering orders were identified. The total energy loss was modeled by summing over the scattering order. The path loss predicted using only the contribution from the first-order agreed with Reilly's analytical single scattering model, as illustrated with different optical geometries. Multiple scattering becomes more dominant for some cases, especially longer propagation distance and larger elevation angles. Our model also provided results that are consistent with the Monte Carlo simulation model. Channel characteristics were investigated in detail, including the effects of varying system geometry and the effects of different atmosphere conditions.

Further study will be conducted to develop an analytical NLOS UV channel impulse response model from which our path loss model can be further validated and channel bandwidth can be predicted. An analytically more tractable path loss model is also a future topic of interest, which can enable more intuitive analysis for the effects of system geometry and parameters.